Question: A monkey is swinging from a tree. On the first swing, she passes through an arc whose length is $24\text{ m}$. With each swing, she travels along an arc that is half as long as the arc of the previous swing. Which expression gives the total length the monkey swings in her first $n$ swings? Choose 1 answer: Choose 1 answer: (Choice A) A $24\cdot \dfrac{(1-1.5^n)}{-0.5}$ (Choice B) B $24\cdot \dfrac{(1-1.5^n)}{0.5}$ (Choice C) C $24\cdot \dfrac{(1-0.5^n)}{-0.5}$ (Choice D) D $24\cdot \dfrac{(1-0.5^n)}{0.5}$
Notice that the lengths of the monkey's swings form a geometric sequence. The total distance traveled after $ n$ swings is the ${\text{sum}}$ of the first $n$ terms in the sequence. This is called a geometric series. This is the formula for that sum: $ S={a}\left(\dfrac{1-{r}^{ n}}{1-{r}}\right)$ where ${a}$ is the first term and ${r}$ is the common ratio. We can use this formula, along with the given information, to find the expression for the sum, $ S$. Using the given information We are given that the length of the first swing is ${24}\text{ m}$. This is the first term $ a$. We are given that each swing is ${0.5}$ times the length of the previous swing (or half as long). This is the common ratio $ r$. We are interested in the first ${n}$ swings, so the number of terms is $ {n}$. We want an expression for the total length the monkey swings. This is the sum $ S$. Writing the sum $ S={24} \cdot \dfrac{1-\left({0.5}\right)^{{n}}}{1-\left({0.5}\right)}$ Answer The total length the monkey swings in her first $n$ swings is: $24\cdot \dfrac{(1-0.5^n)}{0.5}$